Algebra: Square Roots

On this page we hope to clear up problems that you might have with square roots and their uses. Square roots are something you'll never get away from.&nsp; Reassuring, aren't we? :-) Read on or follow the links below to start understanding square roots better!

Simplification of square roots
Addition and subtraction of square roots
Multiplication of square roots
Quiz on square roots

Simplification of Square Roots

When simplifying square roots, keep the Product of Square Roots Theorem in mind. It is outlined below.

If m and n are not negative and are real numbers, then SQRT(m) * SQRT(n) = SQRT(mn).

Examples:

1. Simplify: SQRT(50)
   Solution: Write 50 as a product of prime factors.

             SQRT(5 * 5 * 2)
             
             Use the Product of Square Roots Theorem to
             write the square root of above as a product of
             square roots.

             SQRT(5) * SQRT(5) * SQRT(2)

             By the definition of square roots,
             SQRT(5) * SQRT(5) = 5, so you know have
             5(SQRT(2)).  That is the answer!

2. Simplify: SQRT(147)
   Solution: Write 147 as a product of prime factors.

             SQRT(3 * 7 * 7)

             Use the Product of Square Roots Theorem to
             rewrite the root shown above.

             SQRT(3) * SQRT(7) * SQRT(7)
       
             Use the definition of square roots to simplify
             even further.  The answer is 7(SQRT(3)).

Addition and Subtraction of Square Roots

Adding and subtracting square roots is just like combining like terms when you need to do that with algebraic expressions. If the indices (a square root's index is 2, a cube root's index is 3, a 4th root's index is 4, etc.) or the radicands (the expression under the root sign or enclosed by parentheses after SQRT) are the same, you have a like term on your hands.

Example:

1. Add:      (4 * SQRT(2)) - (5 * SQRT(2)) + 12 * SQRT(2))
   Solution: Combine like terms by adding the numerical
             coefficients.

             (4 - 5 + 12) * SQRT(2)

             After the addition, you get the answer!
             11 * SQRT(2)

Many times, such problems will not be given to you with all the terms alike, or even trickier, the terms will only look different!

Example:

1. Simplify: SQRT(18) + SQRT(8)
   Solution: Write each square root as a product of prime
             factors.

             SQRT(2 * 3 * 3) + SQRT(2 * 2 * 2)

             Use the Product of Square Roots Theorem to
             rewrite each square root above.

             SQRT(2) * SQRT(3) * SQRT(3) +
             SQRT(2) * SQRT(2) * SQRT(2)

             Use the definition of square roots to simplify
             even further.

             (3(SQRT(2))) + (2(SQRT(2)))

             Add like terms to get an answer.

             5(SQRT(2))

Multiplication of Square Roots

Instead of using the Prodcut of Square Roots Theorem to separate a root, we can use it in reverse to multiply radicals, as the following example shows: SQRT(2) * SQRT(3) = SQRT(6).

Example:

1. Simplify: 4(SQRT(3)) * 3(SQRT(2))
   Solution: As with anything else, changing the order of
             the factors does not change the problem, so we
             will rearrange the factors so the problem will
             be a little easier to compute.

             4 * 3 * SQRT(3) * SQRT(2)

             Multiply all the factors together to get the
             answer.
             12(SQRT(6))

Take the quiz on square roots. The quiz is very useful for either review or to see if you've really got the topic down.

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